Domination of a Fuzzy Directed Graph

A new domination parameter in a fuzzy digraph is proposed to espouse a contribution in the domain of domination in a fuzzy graph and a directed graph. Let GD*=V,A be a directed simple graph, where V is a finite nonempty set and A=x,y:x,y∈V,x≠y. A fuzzy digraph GD=σD,μD is a pair of two functions σD:V→0,1 and μD:A→0,1, such that μDx,y≤σDx∧σDy, where x,y∈V. An edge μDx,y of a fuzzy digraph is called an effective edge if μDx,y=σDx∧σDy. Let x,y∈V. The vertex σDx dominates σDy in GD if μDx,y is an effective edge. Let S⊆V, u∈V\S, and v∈S. A subset σDS⊆σD is a dominating set of GD if, for every σDu∈σD\σDS, there exists σDv∈σDS, such that σDv dominates σDu. The minimum dominating set of a fuzzy digraph GD is called the domination number of a fuzzy digraph and is denoted by γGD. In this paper, the concept of domination in a fuzzy digraph is introduced, the domination number of a fuzzy digraph is characterized, and the domination number of a fuzzy dipath and a fuzzy dicycle is modeled.

Identifying vital nodes in complex networks based on information entropy, minimum dominating set and distance

Identifying the vital nodes in complex networks is essential in disease transmission control and network attack protection. In this paper, in order to identify the vital nodes, we define a centrality method named EMDC, which is based on information entropy, minimum dominating set (MDS) and the distance between node pairs. This method calculates the local spreading capability (LSC) of node by information entropy and selects that nodes have the largest value of LSC as core nodes by MDS. Then it defines the node’s spreading capability (SC) to use the sum of weighted distances from a node to the core nodes. Finally, the nodes are ranked by considering SC of their neighbors. The key nodes can be further identified in complex networks. In order to verify the effectiveness of this method, key nodes identification simulation experiments are carried out on 11 real networks, Scale-Free (BA) networks and Small-World (WS) networks, respectively. Experimental results show that this method can more effectively identify the influence of nodes in the networks.

The Suppression of Epidemic Spreading Through Minimum Dominating Set

COVID-19 has infected millions of people, with deaths in more than 200 countries. It is therefore essential to understand the dynamic characteristics of the outbreak and to design effective strategies to restrain the large-scale spread of the epidemic. In this paper, we present a novel framework to depress the epidemic spreading, by leveraging the decentralized dissemination of information. The framework is equivalent to finding a special minimum dominating set for a duplex network which is a general dominating set for one layer and a connected dominating set for another layer. Using the spin glass and message passing theory, we present a belief-propagation-guided decimation (BPD) algorithm to construct the special minimum dominating set. As a consequence, we could immediately recognize the epidemic as soon as it appeared, and rapidly immunize the whole network at minimum cost.

Construction for trees without domination critical vertices

<abstract><p>Denote by $ \gamma(G) $ the domination number of graph $ G $. A vertex $ v $ of a graph $ G $ is called <italic>fixed</italic> if $ v $ belongs to every minimum dominating set of $ G $, and bad if $ v $ does not belong to any minimum dominating set of $ G $. A vertex $ v $ of $ G $ is called <italic>critical</italic> if $ \gamma(G-v) &lt; \gamma(G) $. By using these notations of vertices, we give a construction for trees that does not contain critical vertices.</p></abstract>

Barrakuda: A Hybrid Evolutionary Algorithm for Minimum Capacitated Dominating Set Problem

The minimum capacitated dominating set problem is an NP-hard variant of the well-known minimum dominating set problem in undirected graphs. This problem finds applications in the context of clustering and routing in wireless networks. Two algorithms are presented in this work. The first one is an extended version of construct, merge, solve and adapt, while the main contribution is a hybrid between a biased random key genetic algorithm and an exact approach which we labeled Barrakuda. Both algorithms are evaluated on a large set of benchmark instances from the literature. In addition, they are tested on a new, more challenging benchmark set of larger problem instances. In the context of the problem instances from the literature, the performance of our algorithms is very similar. Moreover, both algorithms clearly outperform the best approach from the literature. In contrast, Barrakuda is clearly the best-performing algorithm for the new, more challenging problem instances.